Compute A 75 Chebyshev Interval

Chebyshev'due south Theorem estimates the minimum proportion of observations that autumn inside a specified number of standard deviations from the mean. This theorem applies to a broad range of probability distributions. Chebyshev's Theorem is too known as Chebyshev's Inequality.

If yous have a mean and standard deviation, you might need to know the proportion of values that lie within, say, plus and minus two standard deviations of the mean. If your data follow the normal distribution, that's easy using the Empirical Rule! However, what if y'all don't know the distribution of your information or you know that it doesn't follow the normal distribution? In that case, Chebyshev's Theorem can help you lot out!

In this post, acquire why Chebyshev'southward theorem is valuable and how to use it to solve problems. Additionally, I'll compare the theorem to the Empirical Rule, which serves a similar purpose.

Equation for Chebyshev'due south Theorem

Chebyshev's Theorem helps you determine where nearly of your data autumn within a distribution of values. This theorem provides helpful results when you have only the mean and standard deviation. You do not demand to know the distribution your data follow.

There are two forms of the equation. One determines how close to the mean the information prevarication and the other calculates how far away from the mean they autumn:

Where k equals the number of standard deviations in which you are interested. K must be greater than 1.

Every bit yous can see, information technology's a fairly straightforward equation.

For more data about the mean and standard difference, read my posts about Measures of Primal Trend and Measures of Variability.

Using Chebyshev's Theorem

By entering values for k into the equation, I've created the tabular array beneath that displays proportions for diverse standard deviations.

Standard Deviations	Minimum % inside	Max % outside
${\sqrt {2}}= 1.41$	0.l	0.fifty
1.5	0.56	0.44
2	0.75	0.25
3	0.89	0.eleven
4	0.94	0.06
v	0.96	0.04

For example, if you're interested in a range of 3 standard deviations effectually the mean, Chebyshev's Theorem states that at least 89% of the observations fall inside that range, and no more than than 11% fall outside that range.

A crucial point to notice is that Chebyshev's Theorem produces minimum and maximum proportions. For example, at to the lowest degree 56% of the observations fall inside ane.5 standard deviations, and a maximum of 44% fall outside.

The theorem does non provide verbal answers, but it places limits on the possible proportions. For the example in a higher place, more than 56% of the observations can lie within one.five standard deviations of the mean.

The minimum and maximum proportions arise due to uncertainties about the data'due south distribution. While the theorem is valuable because it applies to all distributions, information technology also limits the precision of the results.

Example Problems

Suppose you know a dataset has a hateful of 100 and a standard deviation of 10, and you're interested in a range of ± 2 standard deviations. Two standard deviations equal 2 X 10 = xx. Consequently, Chebyshev's Theorem tells you that at least 75% of the values fall between 100 ± 20, equating to a range of 80 – 120. Conversely, no more than 25% autumn outside that range.

An interesting range is ± 1.41 standard deviations. With that range, y'all know that at least half the observations fall within it, and no more than half fall outside of it. If we employ a mean of 100 and a standard departure of 10 again, 1.41 standard deviations equal 14.1. Hence, at least half the values prevarication in the range 100 ± 14.1, or 85.ix – 114.1.

Suppose a form takes a test. The average score is 75 and the standard deviation is 5. What is the proportion of scores that fall between 65 and 85?

The mean is 75. 65 is 10 points below the hateful and 85 is x points higher up the mean. The standard deviation is five. Consequently, you want to determine the proportion of scores that autumn within 10 / 5 = ii standard deviations of the hateful. Using the table to a higher place, you know that at to the lowest degree 75% of the scores will fall within the range of 65 – 85.

Chebyshev's Theorem compared to The Empirical Rule

The Empirical Rule also describes the proportion of information that fall inside a specified number of standard deviations from the mean. However, in that location are several crucial differences between Chebyshev's Theorem and the Empirical Rule.

Chebyshev's Theorem applies to all probability distributions where you can calculate the mean and standard difference. On the other paw, the Empirical Dominion applies only to the normal distribution.

As you saw to a higher place, Chebyshev's Theorem provides approximations. Conversely, the Empirical Rule provides verbal answers for the proportions because the data are known to follow the normal distribution.

Related post: Identifying the Distribution of Your Data

The table beneath compares the results from both methods for the proportions of data falling within the specified number of standard deviations.

Standard Deviations	Empirical Rule	Chebyshev'south Theorem
one	68%	NA
2	95%	≥75%
iii	99.7%	≥88.9%

Again, discover that the Empirical Rule provides exact answers while Chebyshev'southward Theorem gives approximations.

If you know that your data follow the normal distribution, use the Empirical Dominion. Otherwise, Chebyshev'due south Theorem might exist your best choice!

For more than data, read my postal service, Empirical Rule: Definition, Formula, and Uses.